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Statistic identifier: St000145
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Collection: Integer partitions
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Description: The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
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References: [1] Dyson, F. J. Some guesses in the theory of partitions [[MathSciNet:3077150]]
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Code:
def statistic(L):
return L[0] - len(L)
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Statistic values:
[1] => 0
[2] => 1
[1,1] => -1
[3] => 2
[2,1] => 0
[1,1,1] => -2
[4] => 3
[3,1] => 1
[2,2] => 0
[2,1,1] => -1
[1,1,1,1] => -3
[5] => 4
[4,1] => 2
[3,2] => 1
[3,1,1] => 0
[2,2,1] => -1
[2,1,1,1] => -2
[1,1,1,1,1] => -4
[6] => 5
[5,1] => 3
[4,2] => 2
[4,1,1] => 1
[3,3] => 1
[3,2,1] => 0
[3,1,1,1] => -1
[2,2,2] => -1
[2,2,1,1] => -2
[2,1,1,1,1] => -3
[1,1,1,1,1,1] => -5
[7] => 6
[6,1] => 4
[5,2] => 3
[5,1,1] => 2
[4,3] => 2
[4,2,1] => 1
[4,1,1,1] => 0
[3,3,1] => 0
[3,2,2] => 0
[3,2,1,1] => -1
[3,1,1,1,1] => -2
[2,2,2,1] => -2
[2,2,1,1,1] => -3
[2,1,1,1,1,1] => -4
[1,1,1,1,1,1,1] => -6
[8] => 7
[7,1] => 5
[6,2] => 4
[6,1,1] => 3
[5,3] => 3
[5,2,1] => 2
[5,1,1,1] => 1
[4,4] => 2
[4,3,1] => 1
[4,2,2] => 1
[4,2,1,1] => 0
[4,1,1,1,1] => -1
[3,3,2] => 0
[3,3,1,1] => -1
[3,2,2,1] => -1
[3,2,1,1,1] => -2
[3,1,1,1,1,1] => -3
[2,2,2,2] => -2
[2,2,2,1,1] => -3
[2,2,1,1,1,1] => -4
[2,1,1,1,1,1,1] => -5
[1,1,1,1,1,1,1,1] => -7
[9] => 8
[8,1] => 6
[7,2] => 5
[7,1,1] => 4
[6,3] => 4
[6,2,1] => 3
[6,1,1,1] => 2
[5,4] => 3
[5,3,1] => 2
[5,2,2] => 2
[5,2,1,1] => 1
[5,1,1,1,1] => 0
[4,4,1] => 1
[4,3,2] => 1
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => -1
[4,1,1,1,1,1] => -2
[3,3,3] => 0
[3,3,2,1] => -1
[3,3,1,1,1] => -2
[3,2,2,2] => -1
[3,2,2,1,1] => -2
[3,2,1,1,1,1] => -3
[3,1,1,1,1,1,1] => -4
[2,2,2,2,1] => -3
[2,2,2,1,1,1] => -4
[2,2,1,1,1,1,1] => -5
[2,1,1,1,1,1,1,1] => -6
[1,1,1,1,1,1,1,1,1] => -8
[10] => 9
[9,1] => 7
[8,2] => 6
[8,1,1] => 5
[7,3] => 5
[7,2,1] => 4
[7,1,1,1] => 3
[6,4] => 4
[6,3,1] => 3
[6,2,2] => 3
[6,2,1,1] => 2
[6,1,1,1,1] => 1
[5,5] => 3
[5,4,1] => 2
[5,3,2] => 2
[5,3,1,1] => 1
[5,2,2,1] => 1
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => -1
[4,4,2] => 1
[4,4,1,1] => 0
[4,3,3] => 1
[4,3,2,1] => 0
[4,3,1,1,1] => -1
[4,2,2,2] => 0
[4,2,2,1,1] => -1
[4,2,1,1,1,1] => -2
[4,1,1,1,1,1,1] => -3
[3,3,3,1] => -1
[3,3,2,2] => -1
[3,3,2,1,1] => -2
[3,3,1,1,1,1] => -3
[3,2,2,2,1] => -2
[3,2,2,1,1,1] => -3
[3,2,1,1,1,1,1] => -4
[3,1,1,1,1,1,1,1] => -5
[2,2,2,2,2] => -3
[2,2,2,2,1,1] => -4
[2,2,2,1,1,1,1] => -5
[2,2,1,1,1,1,1,1] => -6
[2,1,1,1,1,1,1,1,1] => -7
[1,1,1,1,1,1,1,1,1,1] => -9
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Created: Jul 03, 2013 at 14:34 by Olivier Mallet
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Last Updated: May 29, 2015 at 16:57 by Martin Rubey